| From 1876 to 1878 Lucas developed his theory of the functions Vn, and Un, which now bear his name. He was particularly interested in how these functions could be employed in proving the primality of certain large integers, and as part of his investigations succeeded in demonstrating that the Mersenne number 2127 -- 1 is a prime. Vn and Un can be expressed in terms of the nth powers of the zeros of a quadratic polynomial, and throughout his writings Lucas speculated about the possible extension of these functions to those which could be expressed in terms of the nth powers of the zeros of a cubic polynomial. Indeed, at the end of his life he stated that "by searching for the addition formulas of the numerical functions which originate from recurrence sequences of the third or fourth degree, and by studying in a general way the laws of residues of these functions for prime moduli ... we would arrive at important new properties of prime numbers.";In this thesis we discuss a pair of functions that are easily expressed as certain symmetric polynomials of the zeros of a cubic polynomial and were undoubtedly known to Lucas. We show how their properties seem to underlie the theory that Lucas was seeking. We do this by deriving a number of results which show how the combinatorial and arithmetic aspects of these functions provide an extension of Lucas' theory. Furthermore, we develop many new results, which illustrate the striking analogy between our functions and those of Lucas. We also argue that, while Lucas very likely never developed this theory, it was certainly within his abilities to do so. |
